Algebra Examples

$d_{x} (9 x^{- \frac{1}{2}} + \frac{2}{x^{\frac{5}{2}}})$

Step 1

Since $d_{x}$ is constant with respect to $x$ , the derivative of $d_{x} (9 x^{- \frac{1}{2}} + \frac{2}{x^{\frac{5}{2}}})$ with respect to $x$ is $d_{x} \frac{d}{d x} [9 x^{- \frac{1}{2}} + \frac{2}{x^{\frac{5}{2}}}]$ .

$d_{x} \frac{d}{d x} [9 x^{- \frac{1}{2}} + \frac{2}{x^{\frac{5}{2}}}]$

Step 2

By the Sum Rule, the derivative of $9 x^{- \frac{1}{2}} + \frac{2}{x^{\frac{5}{2}}}$ with respect to $x$ is $\frac{d}{d x} [9 x^{- \frac{1}{2}}] + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}]$ .

$d_{x} (\frac{d}{d x} [9 x^{- \frac{1}{2}}] + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 3

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{- \frac{1}{2}}$ with respect to $x$ is $9 \frac{d}{d x} [x^{- \frac{1}{2}}]$ .

$d_{x} (9 \frac{d}{d x} [x^{- \frac{1}{2}}] + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{1}{2}$ .

$d_{x} (9 (- \frac{1}{2} x^{- \frac{1}{2} - 1}) + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$d_{x} (9 (- \frac{1}{2} x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 6

Combine $- 1$ and $\frac{2}{2}$ .

$d_{x} (9 (- \frac{1}{2} x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 7

Combine the numerators over the common denominator.

$d_{x} (9 (- \frac{1}{2} x^{\frac{- 1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$d_{x} (9 (- \frac{1}{2} x^{\frac{- 1 - 2}{2}}) + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 8.2

Subtract $2$ from $- 1$ .

$d_{x} (9 (- \frac{1}{2} x^{\frac{- 3}{2}}) + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 9

Combine fractions.

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Step 9.1

Move the negative in front of the fraction.

$d_{x} (9 (- \frac{1}{2} x^{- \frac{3}{2}}) + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 9.2

Combine $x^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$d_{x} (9 (- \frac{x^{- \frac{3}{2}}}{2}) + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 9.3

Multiply $- 1$ by $9$ .

$d_{x} (- 9 \frac{x^{- \frac{3}{2}}}{2} + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 9.4

Combine $- 9$ and $\frac{x^{- \frac{3}{2}}}{2}$ .

$d_{x} (\frac{- 9 x^{- \frac{3}{2}}}{2} + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 9.5

Simplify the expression.

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Step 9.5.1

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$d_{x} (\frac{- 9}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 9.5.2

Move the negative in front of the fraction.

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [\frac{2}{x^{\frac{5}{2}}}])$

Step 10

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x^{\frac{5}{2}}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x^{\frac{5}{2}}}]$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 \frac{d}{d x} [\frac{1}{x^{\frac{5}{2}}}])$

Step 11

Apply basic rules of exponents.

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Step 11.1

Rewrite $\frac{1}{x^{\frac{5}{2}}}$ as ${(x^{\frac{5}{2}})}^{- 1}$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 \frac{d}{d x} [{(x^{\frac{5}{2}})}^{- 1}])$

Step 11.2

Multiply the exponents in ${(x^{\frac{5}{2}})}^{- 1}$ .

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Step 11.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 \frac{d}{d x} [x^{\frac{5}{2} \cdot - 1}])$

Step 11.2.2

Multiply $\frac{5}{2} \cdot - 1$ .

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Step 11.2.2.1

Combine $\frac{5}{2}$ and $- 1$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 \frac{d}{d x} [x^{\frac{5 \cdot - 1}{2}}])$

Step 11.2.2.2

Multiply $5$ by $- 1$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 \frac{d}{d x} [x^{\frac{- 5}{2}}])$

Step 11.2.3

Move the negative in front of the fraction.

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 \frac{d}{d x} [x^{- \frac{5}{2}}])$

Step 12

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{5}{2}$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 (- \frac{5}{2} x^{- \frac{5}{2} - 1}))$

Step 13

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 (- \frac{5}{2} x^{- \frac{5}{2} - 1 \cdot \frac{2}{2}}))$

Step 14

Combine $- 1$ and $\frac{2}{2}$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 (- \frac{5}{2} x^{- \frac{5}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 15

Combine the numerators over the common denominator.

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 (- \frac{5}{2} x^{\frac{- 5 - 1 \cdot 2}{2}}))$

Step 16

Simplify the numerator.

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Step 16.1

Multiply $- 1$ by $2$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 (- \frac{5}{2} x^{\frac{- 5 - 2}{2}}))$

Step 16.2

Subtract $2$ from $- 5$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 (- \frac{5}{2} x^{\frac{- 7}{2}}))$

Step 17

Move the negative in front of the fraction.

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 (- \frac{5}{2} x^{- \frac{7}{2}}))$

Step 18

Combine $x^{- \frac{7}{2}}$ and $\frac{5}{2}$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + 2 (- \frac{x^{- \frac{7}{2}} \cdot 5}{2}))$

Step 19

Multiply $- 1$ by $2$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} - 2 \frac{x^{- \frac{7}{2}} \cdot 5}{2})$

Step 20

Combine $- 2$ and $\frac{x^{- \frac{7}{2}} \cdot 5}{2}$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + \frac{- 2 (x^{- \frac{7}{2}} \cdot 5)}{2})$

Step 21

Multiply.

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Step 21.1

Multiply $5$ by $- 2$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + \frac{- 10 x^{- \frac{7}{2}}}{2})$

Step 21.2

Move $x^{- \frac{7}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + \frac{- 10}{2 x^{\frac{7}{2}}})$

Step 22

Factor $2$ out of $- 10$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + \frac{2 \cdot - 5}{2 x^{\frac{7}{2}}})$

Step 23

Cancel the common factors.

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Step 23.1

Factor $2$ out of $2 x^{\frac{7}{2}}$ .

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + \frac{2 \cdot - 5}{2 (x^{\frac{7}{2}})})$

Step 23.2

Cancel the common factor.

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + \frac{2 \cdot - 5}{2 x^{\frac{7}{2}}})$

Step 23.3

Rewrite the expression.

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} + \frac{- 5}{x^{\frac{7}{2}}})$

Step 24

Move the negative in front of the fraction.

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}} - \frac{5}{x^{\frac{7}{2}}})$

Step 25

Simplify.

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Step 25.1

Apply the distributive property.

$d_{x} (- \frac{9}{2 x^{\frac{3}{2}}}) + d_{x} (- \frac{5}{x^{\frac{7}{2}}})$

Step 25.2

Combine terms.

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Step 25.2.1

Combine $d_{x}$ and $\frac{9}{2 x^{\frac{3}{2}}}$ .

$- \frac{d_{x} \cdot 9}{2 x^{\frac{3}{2}}} + d_{x} (- \frac{5}{x^{\frac{7}{2}}})$

Step 25.2.2

Move $9$ to the left of $d_{x}$ .

$- \frac{9 \cdot d_{x}}{2 x^{\frac{3}{2}}} + d_{x} (- \frac{5}{x^{\frac{7}{2}}})$

Step 25.2.3

Combine $d_{x}$ and $\frac{5}{x^{\frac{7}{2}}}$ .

$- \frac{9 d_{x}}{2 x^{\frac{3}{2}}} - \frac{d_{x} \cdot 5}{x^{\frac{7}{2}}}$

Step 25.2.4

Move $5$ to the left of $d_{x}$ .

$- \frac{9 d_{x}}{2 x^{\frac{3}{2}}} - \frac{5 d_{x}}{x^{\frac{7}{2}}}$